In some ways that’s always been the difference between mathematicians and engineers. That’s why engineers, scientists, applied mathematicians, et al. can get away with knowing a little math, reducing practical problems to their sticking points, then looking up the solution to the reduced form in a treatise written two hundred years ago by an abstract mathematician who just loved numbers and never saw the practical side of anything.

Eh, scientists are all skeptics, but as Bertrand Russell (wasn’t he into math?) said,

“Neither acquiescence in skepticism nor acquiescence in dogma is what education should produce. What it should produce is a belief that knowledge is attainable in a measure, though with difficulty; that much of what passes for knowledge at any given time is likely to be more or less mistaken, but that the mistakes can be rectified by care and industry. In acting upon our beliefs, we should be very cautious where a small error would mean disaster; nevertheless it is upon our beliefs that we must act. . . . Knowledge, like other good things, is difficult, but not impossible; the dogmatist forgets the difficulty, the skeptic denies the possibility. Both are mistaken, and their errors, when widespread, produce social disaster.”

I’m sure we’re on the same page with this, but I think it is relevant, for example, to the “controversy” surrounding the computer modeling of the Intergovernmental Panel on Climate Change.

Yes, but Bertrand Russell was also a master of the diabolical counterexample. Just ask Frege.

Frege had this really beautiful, totally intuitive set theory, which he used as a foundation for predicate calculus and arithmetic and all of mathematics.

The problem was with Frege’s concept of the extension of a set (the objects that satisfy a property that the set defines), and the formal way that Frege defined this concept. It permitted a certain kind of meta-reasoning that led to Russell’s Paradox, which goes something like this.

Some sets include themselves as members, while some don’t. (Let’s say, an ordered set of all ordered sets.) We can define the set S of all sets that don’t include themselves as members. Does S include itself or not? Whether it does or doesn’t, the set’s extension contradicts itself.

There have been lots of attempts to try to reduce mathematics to a closed formal procedure. They’ve all turned out to be deeply, but hardly ever obviously, flawed. That’s what mathematical skepticism is about.

Okay, I concede that the power of the counterexample is greater in mathematics than in softer science. But does that show that mathematicians are more skeptical, or does it just reflect that the field is more controlled?

I appreciate the philosophy of the Russell quote, which recognizes that in the real world a single counterexample may not be enough to discount a working theory, especially if there is no alternative or better theory. Essentially, skepticism serves all rational people, but in a world that is chaotic enough to be practically stochastic, we cannot ally our skepticism with rigid formalism; it must instead be tempered with courage and practicality.

But sorry for all the navel-gazing. In the context of the Wolfram post, there’s another point: in mathematics, it can sometimes be proven that a single counterexample is the only counterexample in the space of possibilities, which is another useful result we don’t get much in softer science.